Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 2.7s

Alternatives: 5

Speedup: 1.0×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma (* b 4.0) b (+ (pow (hypot a b) 4.0) -1.0)))

C

double code(double a, double b) {
	return fma((b * 4.0), b, (pow(hypot(a, b), 4.0) + -1.0));
}

Julia

function code(a, b)
	return fma(Float64(b * 4.0), b, Float64((hypot(a, b) ^ 4.0) + -1.0))
end

Wolfram

code[a_, b_] := N[(N[(b * 4.0), $MachinePrecision] * b + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. +-commutative 99.9%

      \[\leadsto \color{blue}{\left(4 \cdot \left(b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right)} - 1 \]

   2. associate--l+ 99.9%

      \[\leadsto \color{blue}{4 \cdot \left(b \cdot b\right) + \left({\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)} \]

   3. add-exp-log 98.9%

      \[\leadsto 4 \cdot \left(b \cdot b\right) + \left(\color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)}} - 1\right) \]

   4. expm1-udef 98.9%

      \[\leadsto 4 \cdot \left(b \cdot b\right) + \color{blue}{\mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)} \]

   5. associate-*r* 98.9%

      \[\leadsto \color{blue}{\left(4 \cdot b\right) \cdot b} + \mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right) \]

   6. fma-def 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot b, b, \mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)\right)} \]

   7. *-commutative 98.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot 4}, b, \mathsf{expm1}\left(\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)\right) \]

   8. expm1-udef 98.9%

      \[\leadsto \mathsf{fma}\left(b \cdot 4, b, \color{blue}{e^{\log \left({\left(a \cdot a + b \cdot b\right)}^{2}\right)} - 1}\right) \]

   9. add-exp-log 99.9%

      \[\leadsto \mathsf{fma}\left(b \cdot 4, b, \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}} - 1\right) \]

   10. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(b \cdot 4, b, {\left(a \cdot a + b \cdot b\right)}^{\color{blue}{\left(\frac{4}{2}\right)}} - 1\right) \]

   11. sqrt-pow2 100.0%

       \[\leadsto \mathsf{fma}\left(b \cdot 4, b, \color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{4}} - 1\right) \]

   12. hypot-udef 100.0%

       \[\leadsto \mathsf{fma}\left(b \cdot 4, b, {\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{4} - 1\right) \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} - 1\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + -1\right) \]

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (pow (hypot a b) 4.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	return pow(hypot(a, b), 4.0) + fma((b * b), 4.0, -1.0);
}

Julia

function code(a, b)
	return Float64((hypot(a, b) ^ 4.0) + fma(Float64(b * b), 4.0, -1.0))
end

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. unpow2 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. unpow1 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   5. associate-*r* 99.9%

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

4. Final simplification 100.0%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (+ (pow (+ (* b b) (* a a)) 2.0) (* 4.0 (* b b))) -1.0))

C

double code(double a, double b) {
	return (pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b))) + -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b))) + -1.0;
}

Python

def code(a, b):
	return (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b))) + -1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(b * b))) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((b * b) + (a * a)) ^ 2.0) + (4.0 * (b * b))) + -1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

2. Final simplification 99.9%

   \[\leadsto \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1 \]

Alternative 4: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{+32}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7e+32) (+ (pow b 4.0) -1.0) (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7e+32) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 7d+32) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7e+32) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7e+32:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7e+32)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7e+32)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7e+32], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.0000000000000002e32

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. unpow2 99.9%

         \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      3. unpow1 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      4. sqr-pow 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

   4. Taylor expanded in b around 0 99.0%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{-1} \]

   5. Taylor expanded in a around 0 82.5%

      \[\leadsto \color{blue}{{b}^{4}} + -1 \]

   if 7.0000000000000002e32 < a

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   2. Taylor expanded in b around 0 93.6%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{+32}:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 5: 69.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -1 + {b}^{4} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (pow b 4.0)))

C

double code(double a, double b) {
	return -1.0 + pow(b, 4.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + (b ** 4.0d0)
end function

Java

public static double code(double a, double b) {
	return -1.0 + Math.pow(b, 4.0);
}

Python

def code(a, b):
	return -1.0 + math.pow(b, 4.0)

Julia

function code(a, b)
	return Float64(-1.0 + (b ^ 4.0))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + (b ^ 4.0);
end

Wolfram

code[a_, b_] := N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. unpow2 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. unpow1 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   5. associate-*r* 99.9%

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

4. Taylor expanded in b around 0 99.2%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{-1} \]

5. Taylor expanded in a around 0 72.2%

   \[\leadsto \color{blue}{{b}^{4}} + -1 \]

6. Final simplification 72.2%

   \[\leadsto -1 + {b}^{4} \]

Reproduce

herbie shell --seed 2023306 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))